PHYSICAL QUATERNIONS
quaternions, obtained by their addition and multiplication, ca: used for relativistic purposes, that is to say, for writing c equations which will satisfy the principle of relativity?
We need not dwell upon the sum a + b + d+ . . . (or ac + bc 4- dc + which is again a physical quaternion, in the original sense of word, and as such, is relativistically available. But having menti< the sum at all, it may be good to observe that a sum of antiva quaternions,* as, for example,
cannot be used. For not only is this sum not covariant with q. with qC) but, when subjected to the Lorentz transformation, split, the two addends being torn asunder, thus
In other words, such a sum is not transferred as a whole from legitimate system of reference to another.
Now for the product of physical quaternions. Begin with simplest case of two factors. Leave aside ah which is split ir act of transformation, thus
dl/^QaQ^Q,
and pass straight on to the product of antivariant f actors , say,
H^aJ>.
Pass from the system S to any other legitimate system S'. '\ H' ~ QcaGQc • Q&Qi whence, by the associative property, and membering that @cQ=i,
Thus, the new quaternion H> though it is neither covariant with standard q nor covariant with q^ is transformed as a whole (< posed of constituents already admitted) and can therefore be for relativistic purposes. A moment's reflection will convince reader that such a procedure will not infringe against the prim of relativity. And the meaning of these abstract remarks
*Any two quaternions of the set